
By choosing an appropriate small \(\alpha \in (0,1/3)\)  we could achieve in Lemma REF  a better threshold. Namely, we would get that it is enough to take \(N\ge C_k d^{(1+\delta )d}\)  for some \(C_k>0\)  and small \(\delta >0\) . This would imply that for a large \(k\in \mathbb {N}\)  and \(d\ge 2^{k+1}\)  it is enough to take \(N\ge C_k d^{(1+\delta )d}\)  in Theorem REF . It is still a growth of order \(d^d\,\)  instead of the expected by us polynomial growth \(d^{k^j}\) , for some \(j\in \mathbb {N}\) . It seems that our approximation method in Lemma REF  is not good enough to achieve a lower order of the threshold for \(N\) . In the case of \(k=2\)  we know that the formula (REF ) holds for \(d\ge 16\)  and \(N\gtrsim d^3\) . Is that the optimal threshold for \(N\) ? What about the other values of \(k\) ?

Denote by \(G_{\rm multi}(k)\)  the smallest integer such that for \(d\ge G_{\rm multi}(k)\)  the multiplicative formula (REF ) holds. What is the optimal value of \(G_{\rm multi}(k)\) ? Our theorem provides the bound \(G_{\rm multi}(k)\le 2^{k+1}\)  with the threshold \(N\gtrsim _k d^{20d}\) . By a careful analysis of our proof one can note that for large \(k\in \mathbb {N}\)  it is enough to take \(G_{\rm multi}(k)\le k(k+1)\)  for \(N\gtrsim _k d^{d^2}\) . Is that trade-off between \(G_{\rm multi}(k)\)  and the threshold for \(N\)  necessary? In the case of the usual additive formula (REF ) it is known, see [1]}, that for large \(k\in \mathbb {N}\)  it holds for \(d\ge G_{\rm add}(k)\)  with
\(G_{\rm add}(k)\le k^2-k+\mathcal {O}(k).\) 
What is the relation between \(G_{\rm multi}(k)\)  and \(G_{\rm add}(k)\) ? It is natural to expect that \(G_{\rm add}(k)\le G_{\rm multi}(k)\)  since we need some place to get rid of the dependence on the dimension. Does the equality \(G_{\rm add}(k)=G_{\rm multi}(k)\)  hold?

Let us consider the generalized Warning problem
\(c_1n_1^k+c_2n_2^k+\ldots +c_dn_d^k=N,\) 
where \(c_1,\ldots ,c_s\)  are given positive integers and \(n_1,\ldots , n_d\)  are arbitrary positive integers. We assume that \(c_1,\ldots ,c_s\)  do not all have a common factor greater than 1. Additionally, in order to ensure solvability, we assume that the congruence
\(c_1n_1^k+c_2n_2^k+\ldots +c_dn_d^k\equiv N\quad (\rm {mod}\,\, p^h)\) 
has a solution for every \(p\in \mathbb {P}\)  and large \(h\in \mathbb {N}\) . For more details see [2]} or [3]}. Denote by \(c=(c_1,c_2,\ldots , c_d)\)  and \(c=\max _{1\le i\le d}c_i\) . The singular series for the generalized Warning problem (REF ) is given by
\(\mathfrak {S}_c(N):=\sum _{q=1}^\infty \sum _{\genfrac{}{}{0.0pt}{}{a=1}{(a, q)=1}}^q\prod _{i=1}^d G(c_ia/q)e\left( -Na/q \right) .\) 
Let \(r_{k}^c(N)\)  denote the number of solutions of the equation (REF ). By following the presented approach and by using some facts from [3]} one can prove the following.
Theorem 3.44 
For each \(k\in \mathbb {N}\)  with \(k\ge 2\)  there exists a constant \(C_{c,k}>0\)  such that, for \(d\ge 2^{k+1}\)  and \(N\ge C_{c,k} d^{20d}\) , we have
\(r_k^{c}(N)=\frac{\mathfrak {S}_c(N)}{(c_1 c_2\cdots c_d)^{1/k}}\frac{\Gamma \left(1+\frac{1}{k}\right)^d}{\Gamma \left(\frac{d}{k}\right)}N^{d/k-1}(1 + o(1)),\) 
where the implicit constants in the symbol \(o(1)\)  depend only on \(c\)  and \(k\) .


